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We study the right and left commutation semigroups of finite metacyclicgroups with trivial centre. These are presented \[G(m,n,k) = \left\langle{a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad(m,n,k\in\mathbb{Z}^+)\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ thesmallest positive integer for which ${k^n} = 1\,\pmod m,$ with the conjugate of$a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \emph{right} and \emph{leftcommutation semigroups of} $G,$ denoted ${\rm P}(G)$ and $\Lambda (G),$ are thesemigroups of mappings generated by $\rho (g):G \to G$ and $\lambda (g):G \toG$ defined by $(x)\rho (g) = [x,g]$ and $(x)\lambda (g) = [g,x],$ where thecommutator of $g$ and $h$ is defined as $[g,h] = {g^{ - 1}}{h^{ - 1}}gh.$ Thispaper builds on a previous study of commutation semigroups of dihedral groupsconducted by the authors with C. Levy. Here we show that a similar approach canbe applied to $G,$ a metacyclic group with trivial centre. We give aconstruction of ${\rm P}(G)$ and $\Lambda (G)$ as unions of \emph{containers},an idea presented in the previous paper on dihedral groups. In the case that$\left\langle a \right\rangle$ is cyclic of order $p$ or ${p^2}$ or its indexis prime, we show that both ${\rm P}(G)$ and $\Lambda (G)$ are disjoint unionsof maximal containers. In these cases, we give an explicit representation ofthe elements of each commutation semigroup as well as formulas for their exactorders. Finally, we extend a result of J. Countryman to show that, for$G(m,n,k)$ with $m$ prime, the condition $\left| {{\rm P}(G)} \right| = \left|{\Lambda (G)} \right|$ is equivalent to ${\rm P}(G) = \Lambda (G).$

Commutation Semigroups of Finite Metacyclic Groups with Trivial Centre